Main references are from the original paper (Matthew Richardson & Pedro Domingos, 2006).

Problem

Traditionally, first-order logic imposes hard constraints on the world. This poses problems in the real world: formulae that may be typically true in the real world are not always true. In most domains, it is difficult to devise non-trivial formulae that are always true. Probabilistic graphical models is a decent solution.

Markov Logic Networks

Markov logic networks relax the hard constraints that first-order logic enforces. When a world violates one formula in a KB, it is less probable, but not impossible. The fewer formulae a world violates, the more probable it is. Each formula is associated with a weight that reflects how strong a constraint it is: the higher the weight, the greater the difference in log probability between a world that satisfies the formula, and one that does not, other things equal.

Formally,

A Markov Logic Network \(L\) is a set of pairs \((F_i, w_i)\), where \(F_i\) is a formula in first-order logic, and \(w_i\) is a real number. Together with a finite set of constants \(C = \left\{ c_1, c-2, \dots, c_{|C|} \right\}\), it defines a Markov Logic Network as follows:

  1. \(M_{L,C}\) contains one binary node for e ach possible grounding of each predicate appearing in \(L\). The binary node takes on value \(1\) if the ground atom is true, and 0 otherwise.
  2. \(M_{L,C}\) contains one feature for each possible grounding of each formula \(F_i\) in \(L\). The value of this feature is \(1\) if the ground formula is true, and 0 otherwise. The weight of the feature is the \(w_i\) associated with \(F_i\) in \(L\).

Bibliography

Richardson, M., & Domingos, P., Markov Logic Networks, Machine Learning, 62(1-2), 107–136 (2006). http://dx.doi.org/10.1007/s10994-006-5833-1